What is the maximum likelihood estimator for $e^{-\theta}( (P(X_i = 0))$?

Question

Suppose $X_i$ is $iid$ $Poisson(\theta)$

What is the maximum likelihood estimator for $e^{-\theta}(= P(Xi = 0))$?

I already found the MLE for the $\theta$. how do you then find the MLE of $e^{-\theta}(= P(Xi = 0))$ ?

Answer 1

By the functional invariance property of the maximum likelihood estimator, the maximum likelihood estimator of $e^{-\theta}$ is just $\color{blue}{e^{-\widehat{\theta}_{MLE}}}$, where $\widehat{\theta}_{MLE}$ is the maximum likelihood estimator of $\theta$.